From my experience and what I would suggest for my own wife...here is my opinion.

1) Where do you suggest staying while looking for a rental: hotel, B & B, AirBNB when in Heredia or San Isidro?

I would stay in San Isidro...you can find many AirBnB and at good prices. Most AirBnB are safe and you can tell how good the renter is by reading the reviews.

2) If you don't want to drive a rent/drive a car, are there services to take you around to see potential rentals without the Gringo pricing kicking in?

YES....I would recommend using an UBER...rates are always better, but sometimes a bit more difficult in smaller towns or non-existant. You can use a taxi, but be careful of the Gringo price.
If you think he is trying to charge to much, you can ask the next taxi and he will do it for cheaper. Or ask someone (a tica) before getting in a taxi what they think would be a correct price for a taxi.

You can even take the BUS almost anywhere and walk to where you need to go.

3) Is it safe to walk around places like Heredia, San Isidro, and other towns during the day time to look for an apartment as a single woman?

In the day, most places are safe. I see single women of all ages walking around Costa Rica. Would I recommend my young daughter...NO! But many here do and it is probably much safer here than in the USA.
In my opinion, I think it is safer in San Isidro and smaller towns than any place in the San Jose Central Valley.  While I am sure there are some safe places in Heredia...everyone KNOWS that the crime is worse in the city, but still not nearly as bad as many places in the USA.

4) Is it easy to find rentals that allow dogs?

YES....Costa Rica is a DOG COUNTRY. Dogs are everywhere!!!  However, some higher end places will not allow it.

5) Are there shelters where you can rescue a dog? Is it expensive to do so?

You will not find tons of shelters like in the USA, but you can find personal shelters. Many Gringso have adopted and created their own shelters and I'm sure you can get a dog for free. (but I have never adopted one...just my guess, dogs are everywhere...you could say...Costa Rica has gone to the dogs. Lol

6) Are furnished rentals easy to find in places like San Isidro? I really don't want to have to "move" a bunch of stuff to CR or when I want to change location when in country.

In San Isidro it is fairly easy. In smaller towns, it can be difficult to find, but not impossible. Takes a bit of looking around. Our tips will help you a lot.

7) How easy/cheap is Uber/Taxis in places like Heredia or San Isidro?

Not expensive...but it all depends on how far you are going. Taxis are commonly used by even the poor Ticos. Many Ticos do NOT have a car, so they go shopping, then pay a taxi to bring them home.

Thanks for posting your topic and I hope this helps you.